Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2178 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | D8 |
| commutator | C8 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C4×D8 |
| prime radical | C7×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C2×C7×C8 ◁ C4×C7×C8 ◁ G | |
146 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,17,14,19,29,18)(10,20,22,31,24,37,23,39)(11,16,25,34,26,35,12,27)(15,30,32,38,33,28,36,21);(8,10)(9,15)(11,21)(12,28)(13,22)(14,24)(16,31)(17,32)(18,36)(19,33)(20,27)(23,29)(25,30)(26,38)(34,37)(35,39);(8,11,13,25,14,26,29,12)(9,16,17,34,19,35,18,27)(10,21,22,30,24,38,23,28)(15,31,32,37,33,39,36,20)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5413);